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Multi-Dimensional Tunneling in Density-Gradient Theory

Multi-Dimensional Tunneling in Density-Gradient Theory. M.G. Ancona Naval Research Laboratory, Washington, DC and K. Lilja Mixed Technology Associates, LLC, Newark, CA. Acknowledgements: MGA thanks ONR for funding support. Introduction.

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Multi-Dimensional Tunneling in Density-Gradient Theory

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  1. Multi-Dimensional Tunneling in Density-Gradient Theory M.G. Ancona Naval Research Laboratory, Washington, DC and K. Lilja Mixed Technology Associates, LLC, Newark, CA Acknowledgements: MGA thanks ONR for funding support.

  2. Introduction • Density-gradient (DG) theory is widely used to analyze quantum confinement effects in devices. • Implemented in commercial codes from Synopsis, Silvaco and ISE. • Similar use of DG theory for tunneling problems has not occurred. Why? • Issues of principle (including is it possible?). • Unclear how to handle multi-dimensions. • Purpose of this talk: DG theory of tunneling and how to apply it in multi-dimensions.

  3. Some Basics • DG theory is a continuumdescription that provides an approximate treatment of quantum transport. • Not microscopic and not equivalent to quantum mechanics so much is lost, e.g., interference, entanglement, Coulomb blockade, etc. • Foundational assumption: The electron and hole gases can be treated as continuous media governed by classical field theory. • Continuum assumption often OK even in ultra-small devices: • Long mean free path doesn't necessarily mean low density. • Long deBroglie l means carrier gases are probability density fluids. • Apparent paradox: How can a classical theory describe quantum transport? A brief answer: DG theory is only macroscopically classical. Hence: • Only macroscopic violations of classical physics must be small. • Material response functions can be quantum mechanical in origin.

  4. where Density-Gradient Theory • DG theory approximates quantum non-locality by making the electron gas equation of state depend on both n and grad(n): • Form of DG equations depends on importance of scattering just as with classical transport:

  5. negligible gen/recomb negligible gen/recomb Density-Gradient(ballistic regime) Density-Gradient (scattering-dominated) Diffusion-Drift Charge/mass conservation: Ballistic Transport inertia negligible inertia negligible Momentum conservation: negligible scattering negligible scattering Electrostatics: These equations describe quantum tunneling through barriers. with with Electron Transport PDEs • General form of PDEs describing macroscopic electron transport:

  6. PDEs for DG Tunneling • Transformations of the DG equations: • Convert from gas pressures to chemical potentials. • Introduce a velocity potential defined by • Governing equations in steady-state: where

  7. Boundary Conditions • Lack of scattering implies infinite mobility plus a lack of mixing of carriers. => Carriers injected from different electrodes must be modeled separately. => Different physics at upstream/downstream contacts represented by different BCs. • Upstream conditions are continuity of • Downstream conditions are continuity of y and Jn plus "tunneling recombination velocity"conditions: where vtrv is a measure of the density of final states.

  8. DG Tunneling in 1D Ancona et al, IEEE Trans. Elect. Dev. (2000) Ancona, Phys. Rev.B (1990) MIM MOS Ancona et al, IEEE Trans. Elect. Dev. (2000) Ancona, unpublished (2002) MOS THBT

  9. DG Tunneling in Multi-D • Test case: STM problem, either a 2D ridge or a 3D tip. • That electrodes are metal implies: • Can ignore band-bending in contacts (ideal metal assumption). • High density means strong gradients and space charge effects. • Goal here is illustration and qualitative behavior, so ignore complexities of metals. • Solve the equations using PROPHET, a powerful PDE solver based on a scripting language (written by Rafferty and Smith at Bell Labs).

  10. Solution Profiles Densities are exponential and current is appropriately concentrated at the STM tip. 2D simulations

  11. I-V Characteristics Current is exponential with strong dependence on curvature. Asymmetrical geometry produces asymmetric I-V as is known to occur in STM. Illustration: Estimate tip convolution --- the loss in STM resolution due to finite radius of curvature.

  12. DG Tunneling in 3-D • Main new issue in 3D is efficiency --- DG approach even more advantageous. • As expected, asymmetry effect even stronger with 3D tip.

  13. Final Remarks • Application of DG theory to MIM tunneling in multi-dimensions has been discussed and illustrated. • Qualitatively the results are encouraging, but quantitatively less sure. • DG confinement reasonably well verified in 1D and multi-D. • Much less work done verifying DG tunneling and all in 1D. • Many interesting problems remain, e.g., gate current in an operating MOSFET. • Main question for the future: Can DG tunneling theory follow DG confinement in becoming an engineering tool? • Need to address theoretical, practical and numerical issues.

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